Consider the context of this answer:

It claims that

$$E[\hat{MSE}_{out}] = E[\hat{MSE}_{in}] + 2/N\sum_{i=1}^NCov(y_i, \hat{y_i})$$

However, the reference (ESLII Section 7.4; PDF page 248) does not prove this. According to this post, $2/N\sum_{i=1}^NCov(y_i, \hat{y_i})$ is actually the difference between in-sample error and the training error. (In the first post, $\hat{MSE}_{out}$ is the out-of-sample estimate of the MSE, which does not appear in section 7.4)

Can we prove the equation in the first post if the model is assumed to be consistent?



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